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% 标题 (title)
%\title{\large  \textbf{Carbuncle Phenomenon and Stability of Upwind Schemes for Two-dimensional Compressible Fluid Flows}}% 文章标题名 (full title name of the article)
%\author{Xin Lei}
%\address{School of Mathematical Sciences, Beijing Normal University, 100875, Beijing, China }
%\author{Jiequan Li}
%\address{Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China}
%\date{\today}
%
%\maketitle  

%\markboth{Xin Lei and Jiequan Li} {Stability of Conservative Schemes}

%\footnotesize


\section{References}

\subsection{Relaxation parameters}
\ 

\cite{ishii_drag_1979}: Magnitude of drag force coefficients (velocity relaxation) in two-phase flow models.


[2001 Saurel]: Revisited BN model and proposed introducing some slight modifications regarding closure relations and major modifications regarding relaxation parameters. 

[2020 Bresch]: The relaxation parameter is linked to the viscosities of the different fluids (which may be small for applications) and the relaxed quantity is linked to the laws chosen at interfaces of the two-fluid system at a mesoscale. 

[2003 Saurel]: For the annular two-phase flow, the analysis determines explicit formulae for the relaxation coefficients as well as the interface variables. 

[2005 BAUDIN]: Develop the second-order explicit relaxation scheme with the computation of new relaxation coefficients and boundary conditions.

\subsection{Comparison of numerical results with(out) pressure relaxation}
\ 

[2007 Munkejord]: Comparison of Roe-type methods for solving the two-fluid model with and without pressure relaxation.

[2010 Munkejord]: Numerical simulations comparing the approaches to relaxation.

[2013 Métayer]: A description of the various relaxation processes is given in the paper.

\cite{ambroso_drift-flux_2008}: The asymptotic behavior of the solutions of barotropic two-phase two-pressure models, with pressure relaxation, drag force and external forces. 


\subsection{Finite-rate relaxation}
\ 

\cite{munkejord_comparison_2007}: The effect of finite-rate pressure relaxation in the Roe5 method (Roe scheme for the 5-equation model) was tested. As the pressure-relaxation parameter was increased, the solution gradually approached that obtained using instantaneous pressure relaxation.
The good correspondence between the results obtained using the Roe4 scheme (Roe scheme for the 4-equation model)  and those of the Roe5 scheme with instantaneous pressure relaxation, indicates that the latter may be regarded as a numerical method to solve the four-equation system.
The Roe5 scheme was found to be significantly more diffusive than the Roe4 scheme.

\cite{ambroso_godunov-type_2012}: Influence of the source terms.

[2020 Pandare]: Choice of finite pressure relaxation time-scale.


\subsection{Stiff relaxation}
\ 

\cite{lund_splitting_2013}: The stiff relaxation term is solved using two approaches: One based on the Backward Euler method, and one using a time-asymptotic scheme. 

[2005 Lallemand]: (Classic paper) Stiff relaxation procedures for multiphase compressible flows.

[2020 Chiocchetti]: Present a technique for constructing robust
solvers for stiff algebraic source terms, such as those typically used for modelling relaxation processes in hyperbolic systems of partial differential equations describing two-phase flows, namely models of the Baer–Nunziato family.


\subsection{Relaxation step}
\

[2012 Liu]: Relaxation step in the computation of water-hammer flows with a two-fluid model.

[2015 Crouzet]: The relaxation step is split into four sub-steps. Each step treats separately velocity relaxation, pressure relaxation, temperature relaxation and mass transfer.

[2012 Hérard]: Emphasis is given on the computation of pressure-velocity-temperature relaxation source terms. Conditions pertaining to the existence and uniqueness of discrete solutions of the relaxation step are given. While focusing on some one-dimensional test cases, the true rates of convergence may be obtained within the evolution step and the relaxation step.

[2004 Gallout]: Relaxation terms are taken into account using a fractional step method.


\subsection{Different models with relaxation}	
\ 

[2006 Trondheim]: The drift-flux model with relaxation procedures.

\cite{hantke_news_2020}: A six-equation model at pressure equilibrium of two ideal gases.
Do not insist that the interfacial pressure tends to the equilibrium pressure when the single-fluid pressures are approaching equilibrium.

[1999 Saurel]: (Classic paper) The relaxation process in the two-phase mixture and the interface conditions.


[2016 Hillairet]: Propose and mathematically derive a generalization of the usual one velocity Baer-Nunziato model with a new relaxation term in the PDE governing the volume fractions. This new relaxation term encodes the change of viscosity and pressure between the different fluids.

[1996 DOWNAR-ZAPOLSKI]: Homogeneous relaxation model (HRM).

[2012 Lund]: A hierarchy of relaxation two-phase flow models is considered, formulated as hyperbolic relaxation systems with source terms. The relaxation terms cause volume, heat, and mass transfer due to differences in pressure, temperature, and chemical potential, respectively, between the two phases. The subcharacteristic condition is a concept closely related to the stability of such relaxation systems. It states that the wave speeds of an equilibrium system never can exceed the speeds of the corresponding relaxation system. 

[2012 Spina]: A single-temperature model for compressible two-phase flow with pressure and velocity relaxations.

\cite{dellacherie_relaxation_2003}: It is possible to construct a class of entropic schemes for the multicomponent Euler system describing a gas or fluid homogeneous mixture at thermodynamic equilibrium by applying a relaxation technique.

[2015 Rodio]: Heat and mass transfer is modeled by applying a thermo-chemical relaxation procedure allowing to deal with metastable states.


\subsection{Reviews}	
\ 

[2017 Saurel]: Modelling compressible dense and dilute two-phase flows.

[2018 Saurel]: Reviews of Compressible Two-Phase Flows.


\subsection{Riemann solvers}
\

[2014 Coquel]: Construct an approximate Riemann solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved
for subsonic relative speeds.


\subsection{Structure of the relaxation zone}
\

[2003 Abgrall]: The wave-like structure one can observe at a macroscopic scale is obtained by homogenisation of such micro-scales problems. The model and method developed in [1999 Saurel] addresses the problem of micro-scale motion by introducing relaxation parameters for pressure and velocity. These relaxation terms are summarising the sum of multidimensional micro-scale motions.

[2001 Kapila]: The structure of the velocity relaxation zone in a hyperbolic, nonconservative, two-phase model is examined in the limit of large drag.

[2016 Schlüter]: Apply fast synchrotron-based X-ray tomography (X-ray CT) to measure the slow relaxation dynamics of fluid interfaces in a glass bead pack after fast drainage of the sample.




%\noindent  \textbf{Acknowledgements}\quad\footnotesize   Thankful words are expressed here.   %致谢与正文间空一行
                                                                                              % (one line between the acknowledgements and the body of the text)
 
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%\footnotesize \noindent\textbf{Appendix A}
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